Arbitrarily Oriented Phase Randomization of Design Ground Motions by Continuous Wavelets
Arbitrarily Oriented Phase Randomization of Design Ground Motions by Continuous Wavelets
Xie, Haoyu;Honda, Riki
2021-10-11 00:00:00
infrastructures Communication Arbitrarily Oriented Phase Randomization of Design Ground Motions by Continuous Wavelets 1 , 2 , 3 Haoyu Xie * and Riki Honda State Key Laboratory of Bridge Engineering Structural Dynamics, CMCT Research & Design Institute Co., Ltd., Chongqing 400067, China School of Civil Engineering, Chongqing University, Chongqing 400045, China Graduate School of Frontier Science, The University of Tokyo, Kashiwa 277-8561, Japan; rikihonda@k.u-tokyo.ac.jp * Correspondence: xhy.civil@yahoo.com Abstract: For dynamic analysis in seismic design, selection of input ground motions is of huge importance. In the presented scheme, complex Continuous Wavelet Transform (CWT) is utilized to simulate stochastic ground motions from historical records of earthquakes with phase disturbance arbitrarily localized in time-frequency domain. The complex arguments of wavelet coefficients are de- termined as phase spectrum and an innovative formulation is constructed to improve computational efficiency of inverse wavelet transform with a pair of random complex arguments introduced and make more candidate wavelets available in the article. The proposed methodology is evaluated by numerical simulations on a two-degree-of-freedom system including spectral analysis and dynamic analysis with Shannon wavelet basis and Gabor wavelet basis. The result shows that the presented scheme enables time-frequency range of disturbance in time-frequency domain arbitrarily oriented and complex Shannon wavelet basis is verified as the optimal candidate mother wavelet for the Citation: Xie, H.; Honda, R. procedure in case of frequency information maintenance with phase perturbation. Arbitrarily Oriented Phase Randomization of Design Ground Keywords: structural seismic design; dynamic analysis; input ground motion; wavelet transform; Motions by Continuous Wavelets. uncertainty Infrastructures 2021, 6, 144. https://doi.org/10.3390/ infrastructures6100144 Academic Editor: Denise-Penelope 1. Introduction N. Kontoni In performance-based earthquake engineering, dynamic analysis is often utilized to evaluate seismic performance of target structures. As a slight fluctuation of input Received: 14 September 2021 ground motion in time-history analysis results in huge difference of nonlinear structural Accepted: 9 October 2021 response [1], uncertainty of design input ground motions should be considered, which Published: 11 October 2021 raises a significant challenge [2,3]. Conventionally, input ground motions for seismic design are accessible from history Publisher’s Note: MDPI stays neutral records of previous earthquakes and artificial ground motion simulation technique by with regard to jurisdictional claims in empirical relationships on fault models [4,5]. Numerical and empirical simulations generate published maps and institutional affil- design ground motions based on considered fault parameters, although specific parameters iations. required for prediction cannot be determined accurately [6]. Meanwhile, due to the complexity of physical models for earthquake phenomenon, there is not a perfect empirical relationship to reproduce the exact ground motion. Indices (intensity measures) based procedures assume that a ground motion which Copyright: © 2021 by the authors. is large enough in terms of indices is supposed to be ‘tough’ for structures. Peak ground Licensee MDPI, Basel, Switzerland. motion acceleration (PGA) is commonly used as an index to quantify the effectiveness This article is an open access article of ground motions from past seismic records, however, considering the fact that the distributed under the terms and number of previous earthquake records by simple amplification cannot meet the request of conditions of the Creative Commons diversity for input ground motions in seismic design, and it is hardly possible that the same Attribution (CC BY) license (https:// earthquake could just happen twice at the particular target site, using ground motion from creativecommons.org/licenses/by/ historical records as design ground motion is restrained. Response spectrum is another 4.0/). Infrastructures 2021, 6, 144. https://doi.org/10.3390/infrastructures6100144 https://www.mdpi.com/journal/infrastructures Infrastructures 2021, 6, 144 2 of 10 index utilized for evaluation of input ground motion [7–10]. In current seismic design codes, acceleration response spectra-compatible ground motion is required for dynamic analysis, although it is not entirely suitable for situations for which nonlinear performance of the structures is dominant because dynamic behavior of nonlinear structures is more sensitive and complicated than it is described by response spectra in frequency domain. As intensity measures cannot fully evaluate the complexity of nonlinear structural performance, another methodology is proposed to fluctuate phase spectrum of an original ground motion and generate artificial ground motions by Fourier Transform (FT), which is called stochastic method [11]. Discrete Wavelet Transform (DWT) is also adopted for time-frequency analysis in similar pattern for the purpose of synthesis of stochastic ground motions [12,13]. The objective of the present paper is to introduce a novel scheme to have time- frequency characteristics of an original signal fluctuated with arbitrary orientation of wavelet phase by using complex continuous wavelet transform and to compare candidates of mother wavelets for optimization of the scheme. Such a design input motion generated for dynamic analysis for seismic design could be considered highly stochastic and uncer- tainty compensated. With the higher diversity and the more information within input ground motions group in dynamic analysis, it is believed a better infrastructure could be built for seismic engineering. 2. Stochastic Method Stochastic method utilizes FT to fluctuate the original ground motion’s Fourier am- plitude spectrum with a random phase spectrum disturbance. Modified ground motion retains some of the parametric and functional descriptions in frequency domain, which is still related to the earthquake magnitude and to the distance from the source. 2.1. Conventional Fourier Analysis iwt The scheme uses sinusoidal function e , basis of Fourier transform, that has shift- invariance and orthogonal properties. The first step of the procedure is to decompose a signal into amplitude spectrum and phase spectrum in frequency domain by discrete Fourier transform, which is given as: N 1 i f t iq( f ) k n k F( f ) = signal(t )e Dt = jF( f )je (1) k å k n=0 where t denotes sampling points of the signal in time domain with sampling duration Dt while f denotes sampling points in frequency domain, and jF( f )j denotes amplitude k k spectrum with q( f ) as phase spectrum of the original signal. After phase spectrum altered, inverse Fourier transform is conducted as: q ( f ) = q( f ) + s( f ) (2) k k k K 1 0 iq ( f ) i f t k k signal (t ) = jF( f )je e D f (3) n å k k=0 where s( f ) denotes artificial phase spectrum aiming at fluctuating the original phase spectrum q( f ). Stochastic method based on discrete Fourier transform has been consid- ered as the most common methodology to generate artificial earthquakes with random phase spectrum. Nevertheless, there is a major demerit of this methodology that the phase distur- bance cannot be localized in time domain as Fourier coefficients could only contain the information of the frequency from the original signal. Infrastructures 2021, 6, 144 3 of 10 2.2. Modification Using DWT Wavelet transform aims to decompose a signal into a set of basis functions consisting of contractions, expansions, and translations of a mother function Y(t, s) [14]. Due to its characteristics of revealing both time and frequency information, wavelet transform has been utilized recently as an alternative to Fourier transform in the process of artificial ground motion simulation. In usual cases, the concept of phase is not rigorously deter- mined [15]. In the scheme, complex arguments of wavelet coefficients are regarded as phase spectrum, and decomposition of the signal by discrete wavelet transform is given as: N 1 1 t k2 iq(j,k) W(j, k) = signal(t )p Y Dt = jW(j, k)je (4) å n n=0 where j represents scale (frequency domain) and k represents transition (time domain), as scale samples of wavelet transforms following a geometric sequence of ratio 2 in dyadic wavelet analysis. Sharing similar manner to Equation (2), phase spectrum in wavelet analysis is altered as: q (j, k) = q(j, k) + s(j, k) (5) where s(j, k) denotes disturbance phase spectrum. The artificial signal is reconstructed by inverse wavelet transform as: N 1 Y(w ) C = Dw (6) jw j n=0 0 1 t k2 D 2 0 1 iq (j,k) j signal (t ) = C D jW(j, k)je p Y D k2 (7) n åå j j 2 4 j k where Y(w ) denotes Fourier transform of wavelet basis Y(t ), and C denotes the opera- n n Y tor for inverse process of continuous wavelet transform. Analytical discrete wavelet transform enables localized disturbances at desired time intervals, by maintaining shift-invariance and orthogonality as Stochastic Method and allows conducting ground motion simulation considering uncertainties in wavelet phase, although time-frequency localization is not arbitrarily oriented due to the characteristics of dyadic distribution of both transition and scale, and the uncertainty principle of time- frequency resolution. 3. Modified Inverse CWT Continuous wavelet transform manages to randomize phase spectrum at any desired area in time-frequency domain, which cannot be realized by discrete wavelet transform as described in Section 2.2. Inverse wavelet transform is well defined by a double integral or sum in transition and scale domain as Equation (7). However, as processing of the method for continuous wavelets requires extremely high performance of the computer and the result precision relies on the certain distribution of sampling scales and shifts, application of the method is often beyond engineers’ capability. Moreover, orthogonality property of wavelet basis is prescribed in conventionally defined inverse transform, so that some wavelet bases with some useful properties like highly compressed product of standard deviations but without orthogonality are unavailable in this research. Considering such problems, we proposed a scheme, by which inverse wavelet transform could be efficiently conducted after phase randomly altering. First, the signal is decomposed continuously in a desired time-frequency domain, which is represented as: Infrastructures 2021, 6, 144 4 of 10 1 t t W s, t = signal t p Y Dt (8) ( ) ( ) where s represents scale (frequency domain) and t represents transition (time domain). Then, the original wavelet coefficients in the certain domain are replaced with coefficients fluctuated by random phase spectrum q(s, t) based on Mersenne Twister. Assuming disturbance as the difference between two sets of wavelet coefficients, it is given as: 1 t t 1 t t n n 0 iq(s,t) dis (t ) = W(s, t)p Y c + W(s, t)e p Y c (9) s,t n s s s s where dis (t ) denotes a single fluctuation on the original signal corresponding to the s,t certain scale and transition (s, t), and c represents a constant coefficient influencing how much this disturbance is amplified. Requirement of computational high performance is avoided by the equation. Equation (9) is derived from: 0 0 dis (t ) = signal(t ) signal (t ) (10) n n n s,t considering for the fluctuation at the certain pair of scale and transition (s, t) in time- frequency domain, from Equation (10) and Equation (7) by decomposing both signal t ( ) and signal (t ) into the sum of wavelet coefficients, we get Equation (9). Since the diversity of total power distribution of the generated signal needs to be enhanced for higher randomness, an extra complex argument is introduced as a modifica- tion to Equation (9), and the artificial ground motion generated from the original signal is given as: h i 1 t t 0 iq (s,t) iq (s,t) 1 2 signal (t ) = signal(t ) W(s, t)p Y e e c (11) n n s t å å s s where q (s, t) and q (s, t) represent two random phase spectra (sets of complex arguments). Equation (11) could be understood as a process that phase spectrum of the original signal is reduced by phase spectrum q (s, t) and then increased with phase spectrum q (s, t). The 1 2 procedure is not linear, so that the pair of random phase spectra in Equation (11) cannot be replaced by single phase spectrum. Shannon wavelet and Gabor wavelet are selected as candidates of mother wavelet in the article due to appropriate qualities for this research. Complex Shannon wavelet is an analytic hardy wavelet with sinc function as its real and imaginary part which has no negative frequency component, so that total power of the signal is maintained in the scheme based on complex Shannon wavelet. Its real and imaginary parts are shown in Figure 1a and the same parts in frequency domain is shown in Figure 1b. Gabor wavelet minimizes the product of its standard deviations in the time and frequency domain, meaning that the uncertainty in information carried by this wavelet is minimized, although total power of the signal may not be maintained in the scheme based on Gabor wavelet. Real and imaginary parts of Gabor wavelet is shown in Figure 2a and the same parts in frequency domain is shown in Figure 2b. Infrastructures 2021, 6, x FOR PEER REVIEW 4 of 10 where 𝑠 represents scale (frequency domain) and 𝜏 represents transition (time domain). Then, the original wavelet coefficients in the certain domain are replaced with coefficients ( ) fluctuated by random phase spectrum 𝜃 𝑠 ,𝜏 based on Mersenne Twister. Assuming dis- turbance as the difference between two sets of wavelet coefficients, it is given as: 1 𝑡 −𝜏 1 𝑡 −𝜏 𝑛 𝑛 ′ (𝑠 ,𝜏 ) ( ) ( ) ( ) 𝑑 𝑡 = −𝑊 𝑠 ,𝜏 𝛹 ( )𝑐 +𝑊 𝑠 ,𝜏 𝑒 𝛹 ( )𝑐 (9) 𝑠 ,𝜏 𝑛 𝑠 𝑠 𝑠 𝑠 √ √ where 𝑠𝑑𝑖 (𝑡 ) denotes a single fluctuation on the original signal corresponding to the 𝑠 ,𝜏 𝑛 ( ) certain scale and transition 𝑠 ,𝜏 , and 𝑐 represents a constant coefficient influencing how much this disturbance is amplified. Requirement of computational high performance is avoided by the equation. Equation (9) is derived from: ′ ′ 𝑠𝑑𝑖 (𝑡 )= 𝑔𝑛𝑖𝑎𝑙𝑠 (𝑡 )−𝑔𝑛𝑖𝑎𝑙𝑠 (𝑡 ) (10) 𝑠 ,𝜏 𝑛 𝑛 𝑛 ( ) considering for the fluctuation at the certain pair of scale and transition 𝑠 ,𝜏 in time- ( ) frequency domain, from Equation (10) and Equation (7) by decomposing both 𝑔𝑛𝑖𝑎𝑙𝑠 𝑡 and 𝑔𝑛𝑖𝑎𝑙𝑠 (𝑡 ) into the sum of wavelet coefficients, we get Equation (9). Since the diversity of total power distribution of the generated signal needs to be enhanced for higher randomness, an extra complex argument is introduced as a modifi- cation to Equation (9), and the artificial ground motion generated from the original signal is given as: 1 𝑡 −𝜏 ′ 𝑖 𝜃 (𝑠 ,𝜏 ) 𝑖 𝜃 (𝑠 ,𝜏 ) 1 2 𝑔𝑛𝑖𝑎𝑙𝑠 (𝑡 )= 𝑔𝑛𝑖𝑎𝑙𝑠 (𝑡 )−∑ ∑ 𝑊 (𝑠 ,𝜏 ) 𝛹 ( )[𝑒 −𝑒 ]𝑐 (11) 𝑛 𝑛 𝑠 𝜏 𝑠 where 𝜃 (𝑠 ,𝜏 ) and 𝜃 (𝑠 ,𝜏 ) represent two random phase spectra (sets of complex argu- 1 2 ments). Equation (11) could be understood as a process that phase spectrum of the original ( ) signal is reduced by phase spectrum 𝜃 𝑠 ,𝜏 and then increased with phase spectrum ( ) 𝜃 𝑠 ,𝜏 . The procedure is not linear, so that the pair of random phase spectra in Equation (11) cannot be replaced by single phase spectrum. Shannon wavelet and Gabor wavelet are selected as candidates of mother wavelet in the article due to appropriate qualities for this research. Complex Shannon wavelet is an analytic hardy wavelet with sinc function as its real and imaginary part which has no negative frequency component, so that total power of the signal is maintained in the scheme based on complex Shannon wavelet. Its real and imaginary parts are shown in Figure 1a and the same parts in frequency domain is shown in Figure 1b. Gabor wavelet minimizes the product of its standard deviations in the time and frequency domain, mean- ing that the uncertainty in information carried by this wavelet is minimized, although total power of the signal may not be maintained in the scheme based on Gabor wavelet. Infrastructures 2021, 6, 144 5 of 10 Real and imaginary parts of Gabor wavelet is shown in Figure 2a and the same parts in frequency domain is shown in Figure 2b. Infrastructures 2021, 6, x FOR PEER REVIEW 5 of 10 (a) (b) Figure 1. (a) Time and (b) frequency characteristics of complex Shannon wavelet basis. Figure 1. (a) Time and (b) frequency characteristics of complex Shannon wavelet basis. (a) (b) Figure 2. (a) Time and (b) frequency characteristics of Gabor wavelet basis. Figure 2. (a) Time and (b) frequency characteristics of Gabor wavelet basis. 4 4.. Num Numerical erical S Simulation imulation 4.1. Spectral Analysis 4.1. Spectral Analysis The EW component of a strong ground motion data of 2000 Tottori Earthquake The EW component of a strong ground motion data of 2000 Tottori Earthquake (Mw (M 6.7) observed from KYT001 station of online database Kiban-Kyoshin-net [16] is se- 6.7) observed from KYT001 station of online database Kiban-Kyoshin-net [16] is selected lected for numerical simulation in the article as it is considered a typical urban earthquake. for numerical simulation in the article as it is considered a typical urban earthquake. Am- Amplitude spectra of the time series by Shannon wavelet transform and Gabor wavelet plitude spectra of the time series by Shannon wavelet transform and Gabor wavelet trans- transform are shown in Figure 3. form are shown in Figure 3. In case of Figure 3, it is verified that both amplitude spectra share similar dominant range for contributing the most of total power of the signal in time-frequency domain (the scale is amplified according to the center frequency of mother wavelets, so that the scale in Shannon wavelet amplitude spectrum is four times the scale in Gabor wavelet amplitude spectrum). Based on the distribution of dominant range of wavelet amplitude spectra, in the article, 15 to 45 s transition and 0 to 3 scale in Shannon wavelet spectrum are selected as the target domain, while 15 to 45 s transition and 0 to 0.75 scale in Gabor wavelet spectrum are selected as the target domain. 10,000 wavelet coefficients in the target domain are chosen randomly to be disturbed by uncertainties of two sets of phase spectra q (s, t) and q (s, t) from 0 to in Equation (10). The disturbances generated by two wavelets based on Equation (9) with the certain set of phase spectra are shown in Figure 4, and response spectra of two artificial ground motions compared to original signal are shown in Figure 5. (a) (b) Figure 3. Amplitude spectra of the time series by (a) Shannon wavelets and (b) Gabor wavelets. In case of Figure 3, it is verified that both amplitude spectra share similar dominant range for contributing the most of total power of the signal in time-frequency domain (the scale is amplified according to the center frequency of mother wavelets, so that the scale in Shannon wavelet amplitude spectrum is four times the scale in Gabor wavelet ampli- tude spectrum). Based on the distribution of dominant range of wavelet amplitude spec- tra, in the article, 15 to 45 s transition and 0 to 3 scale in Shannon wavelet spectrum are selected as the target domain, while 15 to 45 s transition and 0 to 0.75 scale in Gabor wave- let spectrum are selected as the target domain. 10,000 wavelet coefficients in the target domain are chosen randomly to be disturbed by uncertainties of two sets of phase spectra 𝜃 (𝑠 ,𝜏 ) and 𝜃 (𝑠 ,𝜏 ) from 0 to π in Equation (10). The disturbances generated by two 1 2 wavelets based on Equation (9) with the certain set of phase spectra are shown in Figure 𝑖𝑠 𝑖𝜃 Infrastructures 2021, 6, x FOR PEER REVIEW 5 of 10 Figure 1. (a) Time and (b) frequency characteristics of complex Shannon wavelet basis. (a) (b) Figure 2. (a) Time and (b) frequency characteristics of Gabor wavelet basis. 4. Numerical Simulation 4.1. Spectral Analysis The EW component of a strong ground motion data of 2000 Tottori Earthquake (Mw 6.7) observed from KYT001 station of online database Kiban-Kyoshin-net [16] is selected for numerical simulation in the article as it is considered a typical urban earthquake. Am- Infrastructures 2021, 6, 144 6 of 10 plitude spectra of the time series by Shannon wavelet transform and Gabor wavelet trans- form are shown in Figure 3. Infrastructures 2021, 6, x FOR PEER REVIEW 6 of 10 Infrastructures 2021, 6, x FOR PEER REVIEW 6 of 10 4, and response spectra of two artificial ground motions compared to original signal are 4, and response spectra of two artificial ground motions compared to original signal are shown in Figure 5. shown in Figure 5. It can be seen from Figure 4 that the disturbances generated by both wavelet bases It can be seen from Figure 4 that the disturbances generated by both wavelet bases are restrained in the certain time domain (15 to 45 s) as we designed. In contrast, Gabor are restrained in the certain time domain (15 to 45 s) as we designed. In contrast, Gabor wavelet provides higher performance of filtering noise in the domain aside from domi- wavelet provides higher performance of filtering noise in the domain aside from domi- nant time range due to the property of minimization the product of its standard devia- nant time range due to the property of minimization the product of its standard devia- tions. However, in case of Figure 5, it can be seen that only complex Shannon wavelet out tions. However, in case of Figure 5, it can be seen that only complex Shannon wavelet out of two candidates relatively manages to maintain the shape of response spectrum and of two candidates relatively manages to maintain the shape of response spectrum and power distribution of the ground motion in frequency domain, since it is an analytic wave- power distribution of the ground motion in frequency domain, since it is an analytic wave- let. As it is in stochastic method, the amplitude information in frequency domain is con- let. As it is in stochastic metho d, the amplitude information in frequency doma in is con- siderably preserved by Shannon wavelet, while modification in the time domain of accel- siderably preserved by Shannon wavelet, while modification in the time domain of accel- (a) (b) eration series and phase information in frequency domain leads to the diversity enhance- eration series and phase information in frequency domain leads to the diversity enhance- Figure 3. Amplitude spectra of the time series by (a) Shannon wavelets and (b) Gabor wavelets. ment of the seismic responses to the certain ground motion set. Figure 3. Amplitude spectra of the time series by (a) Shannon wavelets and (b) Gabor wavelets. ment of the seismic responses to the certain ground motion set. In case of Figure 3, it is verified that both amplitude spectra share similar dominant range for contributing the most of total power of the signal in time-frequency domain (the scale is amplified according to the center frequency of mother wavelets, so that the scale in Shannon wavelet amplitude spectrum is four times the scale in Gabor wavelet ampli- tude spectrum). Based on the distribution of dominant range of wavelet amplitude spec- tra, in the article, 15 to 45 s transition and 0 to 3 scale in Shannon wavelet spectrum are selected as the target domain, while 15 to 45 s transition and 0 to 0.75 scale in Gabor wave- let spectrum are selected as the target domain. 10,000 wavelet coefficients in the target domain are chosen randomly to be disturbed by uncertainties of two sets of phase spectra 𝜃 (𝑠 ,𝜏 ) and 𝜃 (𝑠 ,𝜏 ) from 0 to π in Equation (10). The disturbances generated by two 1 2 wavelets based on Equation (9) with the certain set of phase spectra are shown in Figure (a) (b) (a) (b) Figure 4. Random disturbance generated by (a) Shannon wavelets and (b) Gabor wavelets. Figure 4. Random disturbance generated by (a) Shannon wavelets and (b) Gabor wavelets. Figure 4. Random disturbance generated by (a) Shannon wavelets and (b) Gabor wavelets. (a) (b) (a) (b) Figure Figure 5 5. . Co Comparison mparison bet between ween respo response nse spect spectra ra of o of rig original inal ground ground motimotion on and a and rtifiartificial cial ground ground motiomotion n genera generated ted by (a) Figure 5. Comparison between response spectra of original ground motion and artificial ground motion generated by (a) Shannon wavelets and (b) Gabor wavelets. by (a) Shannon wavelets and (b) Gabor wavelets. Shannon wavelets and (b) Gabor wavelets. It can be seen from Figure 4 that the disturbances generated by both wavelet bases To distinctly illustrate the improvement by the proposed scheme compared to sto- To distinctly illustrate the improvement by the proposed scheme compared to sto- are restrained in the certain time domain (15 to 45 s) as we designed. In contrast, Gabor chastic method in Section 2, the certain disturbance and artificial ground motion time se- chastic method in Section 2, the certain disturbance and artificial ground motion time se- wavelet provides higher performance of filtering noise in the domain aside from dominant ries modified by stochastic method are shown in Figure 6. Contrast to Figure 4, the dis- ries modified by stochastic method are shown in Figure 6. Contrast to Figure 4, the dis- time range due to the property of minimization the product of its standard deviations. turbance in Figure 6a by stochastic method is not restrained in the dominant time domain. turbance in Figure 6a by stochastic method is not restrained in the dominant time domain. The fluctuation in whole time domain cannot be averted by traditional stochastic method. The fluctuation in whole time domain cannot be averted by traditional stochastic method. Infrastructures 2021, 6, 144 7 of 10 However, in case of Figure 5, it can be seen that only complex Shannon wavelet out of two candidates relatively manages to maintain the shape of response spectrum and power distribution of the ground motion in frequency domain, since it is an analytic wavelet. As it is in stochastic method, the amplitude information in frequency domain is considerably preserved by Shannon wavelet, while modification in the time domain of acceleration series and phase information in frequency domain leads to the diversity enhancement of the seismic responses to the certain ground motion set. To distinctly illustrate the improvement by the proposed scheme compared to stochas- tic method in Section 2, the certain disturbance and artificial ground motion time series modified by stochastic method are shown in Figure 6. Contrast to Figure 4, the disturbance Infrastructures 2021, 6, x FOR PEER REVIEW 7 of 10 Infrastructures 2021, 6, x FOR PEER REVIEW 7 of 10 in Figure 6a by stochastic method is not restrained in the dominant time domain. The fluctuation in whole time domain cannot be averted by traditional stochastic method. (a) (b) (a) (b) Figure 6. A case of artificial ground motion simulation by Stochastic Method: (a) Random disturbance; (b) Original and Figure 6. A case of artificial ground motion simulation by Stochastic Method: (a) Random disturbance; (b) Original and Figure 6. A case of artificial ground motion simulation by Stochastic Method: (a) Random disturbance; (b) Original and artificial time series. artificial time series. artificial time series. 4.2. 4.2. Dynamic Dynamic Analysis Analysis 4.2. Dynamic Analysis Based on the above, complex Shannon wavelet is considered the optimal mother Based on the above, complex Shannon wavelet is considered the optimal mother Based on the above, complex Shannon wavelet is considered the optimal mother wavelet wavelet out out of of two two candidates candidates due due to to its its power power maintenance maintenance with with phase phase pert pertur urbation. bation. wavelet out of two candidates due to its power maintenance with phase perturbation. Dynamic Dynamic analysis analysis i iss to to be be conducted conducted with with complex complex Shannon Shannon wavelet wavelet in inthis this sec section. tion. Dynamic analysis is to be conducted with complex Shannon wavelet in this section. An idealized 2-story shear-frame-model is used here as the target structure. The An idealized 2-story shear-frame-model is used here as the target structure. The An idealized 2-story shear-frame-model is used here as the target structure. The building is simulated by a 2-degree-of-freedom model in which nonlinear behavior of building is simulated by a 2-degree-of-freedom model in which nonlinear behavior of building is simulated by a 2-degree-of-freedom model in which nonlinear behavior of springs is expressed by tri-linear Clough model, modified following a case in Architectural springs is expressed by tri-linear Clough model, modified following a case in Architec- springs is expressed by tri-linear Clough model, modified following a case in Architec- Institute of Japan [17] as shown in Figure 7. The seismic response of the structure is tural Institute of Japan [17] as shown in Figure 7. The seismic response of the structure is tural Institute of Japan [17] as shown in Figure 7. The seismic response of the structure is simulated in software Open System for Earthquake Engineering Simulation [18] by utilizing simulated in software Open System for Earthquake Engineering Simulation [18] by utiliz- simulated in software Open System for Earthquake Engineering Simulation [18] by utiliz- time domain Newmark-b method. ing time domain Newmark-β method. ing time domain Newmark-β method. Mass 1 Mass 1 Nonlinear Spring 1 Nonlinear Spring 1 Mass 2 Mass 2 Nonlinear Spring 2 Nonlinear Spring 2 Figure Figure 7. 7. T T ar arge gettstr struc uctur ture e and andanalysis analysis model modecorr l corr esponded. esponded Figure 7. Target structure and analysis model corresponded Through modal analysis (initial stiffness is considered), modal frequencies and peri- Through modal analysis (initial stiffness is considered), modal frequencies and peri- ods are obtained as shown in Table 1, according to which, 15 to 45 s and 0.4 to 0.6 Hertz ods are obtained as shown in Table 1, according to which, 15 to 45 s and 0.4 to 0.6 Hertz in Shannon wavelet spectrum are selected as the target domain to disturb the certain in Shannon wavelet spectrum are selected as the target domain to disturb the certain phase spectrum of 1000 random wavelet coefficients. In a particular case, the acceleration phase spectrum of 1000 random wavelet coefficients. In a particular case, the acceleration time series of original signal and artificial signal are shown in Figure 8, and the vertical time series of original signal and artificial signal are shown in Figure 8, and the vertical seismic deformation responses of the target structure are shown in Figure 9. seismic deformation responses of the target structure are shown in Figure 9. Table 1. Modal frequencies and periods of the analysis model Table 1. Modal frequencies and periods of the analysis model Frequency (Hz) Period (s) Frequency (Hz) Period (s) 1st Mode 0.39 2.57 1st Mode 0.39 2.57 2nd Mode 0.78 1.29 2nd Mode 0.78 1.29 Infrastructures 2021, 6, 144 8 of 10 Through modal analysis (initial stiffness is considered), modal frequencies and periods are obtained as shown in Table 1, according to which, 15 to 45 s and 0.4 to 0.6 Hertz in Shannon wavelet spectrum are selected as the target domain to disturb the certain phase spectrum of 1000 random wavelet coefficients. In a particular case, the acceleration time series of original signal and artificial signal are shown in Figure 8, and the vertical seismic deformation responses of the target structure are shown in Figure 9. Table 1. Modal frequencies and periods of the analysis model. Frequency (Hz) Period (s) 1st Mode 0.39 2.57 Infrastructures 2021, 6, x FOR PEER REVIEW 8 of 10 2nd Mode 0.78 1.29 Infrastructures 2021, 6, x FOR PEER REVIEW 8 of 10 Figure 8. Acceleration time series of original signal and artificial signal. Figure 8. Acceleration time series of original signal and artificial signal. Figure 8. Acceleration time series of original signal and artificial signal. Figure 9. Vertical seismic responses of deformation of the target structure. Figure 9. Vertical seismic responses of deformation of the target structure. Figure 9. Vertical seismic responses of deformation of the target structure. In case of Figures 8 and 9, it could be found that artificial ground motion generated In case of Figures 8 and 9, it could be found that artificial ground motion generated by In case of Figures 8 and 9, it could be found that artificial ground motion generated by the proposed methodology excites completely different deformative response of the the proposed methodology excites completely different deformative response of the target by the proposed methodology excites completely different deformative response of the target structure, due to the random phase spectrum in the structural dominant time-fre- structure, due to the random phase spectrum in the structural dominant time-frequency range. target structure, due to the random phase spectrum in the structural dominant time-fre- quenc Besides y range. the certain case above, 50 sets of random phase spectra are generated to disturb quency range. the original Besides signal the certa byicontinuous n case above, wavelet 50 sets o transform. f random The phase same spec pr tra ocess are genera is carried ted out to dby is- Besides the certain case above, 50 sets of random phase spectra are generated to dis- turb discr ete the wavelet original transform signal by c as on the tinuo base us wa line.v Since elet tr CWT ansfoarbitrarily rm. The saorients me prophase cess is ca disturbance rried out turb the original signal by continuous wavelet transform. The same process is carried out b in y fr dequency iscrete wdomain, avelet tra the nsfo frrm equencies as the bselected aseline. with SinceCWT CWTcould arbitra be rimor ly ori e ents corelated phase to dimodal sturb- by discrete wavelet transform as the baseline. Since CWT arbitrarily orients phase disturb- frequencies of the target structure, and thus enables the methodology to be more efficient ance in frequency domain, the frequencies selected with CWT could be more corelated to ance in frequency domain, the frequencies selected with CWT could be more corelated to because artificial ground motion synthesized by the scheme is supposed to excite more modal frequencies of the target structure, and thus enables the methodology to be more modal frequencies of the target structure, and thus enables the methodology to be more variant seismic response of the target structure due to resonance. Maximum displacement efficient because artificial ground motion synthesized by the scheme is supposed to excite efficient because artificial ground motion synthesized by the scheme is supposed to excite of the target structure is selected as the intensity measure to evaluate seismic performance. more variant seismic response of the target structure due to resonance. Maximum dis- more variant seismic response of the target structure due to resonance. Maximum dis- The result shows that the mean square error of maximum displacements by DWT scheme is placement of the target structure is selected as the intensity measure to evaluate seismic placement of the target structure is selected as the intensity measure to evaluate seismic 1.97, while the mean square error of maximum displacements by CWT scheme is only 0.489. performance. The result shows that the mean square error of maximum displacements by performance. The result shows that the mean square error of maximum displacements by Figure 10 shows the distribution of two sets of maximum displacements responded to DWT scheme is 1.97, while the mean square error of maximum displacements by CWT DWT scheme is 1.97, while the mean square error of maximum displacements by CWT artificial ground motions representatively disturbed by CWT and DWT. As CWT explicitly scheme is only 0.489. Figure 10 shows the distribution of two sets of maximum displace- scheme is only 0.489. Figure 10 shows the distribution of two sets of maximum displace- amplifies the variance of seismic responses, it is considered that the diversity of input ments responded to artificial ground motions representatively disturbed by CWT and ments responded to artificial ground motions representatively disturbed by CWT and DWT. As CWT explicitly amplifies the variance of seismic responses, it is considered that DWT. As CWT explicitly amplifies the variance of seismic responses, it is considered that the diversity of input ground motions is enhanced and the uncertainty in seismic design the diversity of input ground motions is enhanced and the uncertainty in seismic design by dynamic analysis is decreased with the proposed method. by dynamic analysis is decreased with the proposed method. Infrastructures 2021, 6, 144 9 of 10 Infrastructures 2021, 6, x FOR PEER REVIEW 9 of 10 ground motions is enhanced and the uncertainty in seismic design by dynamic analysis is decreased with the proposed method. (a) (b) Figure 10. Distribution of max response deformations by artificial GMs synthesized using (a) CWT and (b) DWT. Figure 10. Distribution of max response deformations by artificial GMs synthesized using (a) CWT and (b) DWT. 5. Conclusions 5. Conclusions For the synthesis of input artificial ground motions for seismic design, there are For the synthesis of input artificial ground motions for seismic design, there are nu- numerous methodologies based on different theories. In Stochastic Method, Fourier Trans- merous methodologies based on different theories. In Stochastic Method, Fourier Trans- form is utilized to disturb the Fourier phase spectrum randomly while the information form is utilized to disturb the Fourier phase spectrum randomly while the information in in frequency domain and the total power of ground motions are maintained. However, frequency domain and the total power of ground motions are maintained. However, lo- localized disturbances at desired time intervals are not available since sinusoidal base calized disturbances at desired time intervals are not available since sinusoidal base func- function extends in the whole time-domain. By defining complex arguments of the wavelet tion extends in the whole time-domain. By defining complex arguments of the wavelet coefficients as the conception of phase, discrete wavelet transform is introduced to fix such coefficients as the conception of phase, discrete wavelet transform is introduced to fix such problem, although localized disturbances cannot be arbitrarily oriented due to the property problem, although localized disturbances cannot be arbitrarily oriented due to the prop- of dyadic distribution of both transition and scale coefficients. erty of dyadic distribution of both transition and scale coefficients. In the article, a new methodology based on continuous complex wavelet transform In the article, a new methodology based on continuous complex wavelet transform is proposed, by which localization of phase disturbance in time-frequency domain with is proposed, by which localization of phase disturbance in time-frequency domain with arbitrary orientation is realized. Moreover, the computational performance is required at a arbitrary orientation is realized. Moreover, the computational performance is required at lower rate and more candidate wavelets, which is not analytic, bases become available in a lower rate and more candidate wavelets, which is not analytic, bases become available the scheme, enabled by the modified algorithm of inverse wavelet transform. Results of in the scheme, enabled by the modified algorithm of inverse wavelet transform. Results numerical simulation indicate that by utilizing the methodology, random phase disturbance of numerical simulation indicate that by utilizing the methodology, random phase dis- could be precisely localized in the desired time-frequency domain, while variability and turbance could be precisely localized in the desired time-frequency domain, while varia- diversity of input ground motions are considerably enhanced. For the selection of wavelet bility and diversity of input ground motions are considerably enhanced. For the selection candidates, complex Shannon wavelet is evaluated as a better choice for the methodology of wavelet candidates, complex Shannon wavelet is evaluated as a better choice for the since the total power and frequency characteristics are highly maintained with random methodology since the total power and frequency characteristics are highly maintained phase perturbation. with random phase perturbation. For future research, more case studies are supposed to be conducted in order to For future research, more case studies are supposed to be conducted in order to eval- evaluate the proposed scheme explicitly referring to engineering practice. uate the proposed scheme explicitly referring to engineering practice. Author Contributions: Conceptualization, R.H. and H.X.; methodology, H.X. and R.H.; validation, Author Contributions: Conceptualization, R.H. and H.X.; methodology, H.X. and R.H.; validation, H.X.; formal analysis, H.X.; investigation, H.X.; data curation, H.X.; writing—original draft prepara- H.X.; formal analysis, H.X.; investigation, H.X.; data curation, H.X.; writing—original draft prepa- tion, H.X.; writing—review and editing, H.X.; visualization, H.X.; supervision, R.H. All authors have ration, H.X.; writing—review and editing, H.X.; visualization, H.X.; supervision, R.H. All authors read and agreed to the published version of the manuscript. have read and agreed to the published version of the manuscript. Funding: This research was funded by “Major Program of National Natural Science Foundation of Funding: This research was funded by “Major Program of National Natural Science Foundation of China, grant number 12032008” and “National Key Research and Development Program in China, China, grant number 12032008” and “National Key Research and Development Program in China, grant number 2017YFC0806009”. grant number 2017YFC0806009”. Data Availability Statement: Some or all data, models, or code generated or used during the study Data Availability Statement: Some or all data, models, or code generated or used during the study are available from the corresponding author by request. are available from the corresponding author by request. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest. Infrastructures 2021, 6, 144 10 of 10 References 1. Ma, H.; Zhuo, W.; Lavorato, D.; Nuti, C.; Fiorentino, G.; Marano, G.C.; Greco, R.; Briseghella, B. Probabilistic seismic response and uncertainty analysis of continuous bridges under near-fault ground motions. Front. Struct. Civ. Eng. 2019, 13, 1510–1519. [CrossRef] 2. Honda, R. Study for implementation of anti-catastrophe-oriented seismic design. J. Jpn. Assoc. Earthq. Eng. 2018, 37, 1077–1086. (In Japanese) [CrossRef] 3. Jeong, K.H.; Lee, H.S. Ground-motion prediction equation for South Korea based on recent earthquake records. Earthq. Struct. 2018, 15, 29–44. 4. Atkinson, G.M. The interface between empirical and simulation-based ground-motion models. Pure Appl. Geophys. 2018, 177, 2069–2081. [CrossRef] [PubMed] 5. Atkinson, G.M.; Bommer, J.; Abrahamson, N. Alternative approaches to modeling epistemic uncertainty in ground motions in probabilistic seismic hazard analysis. Seismol. Res. Lett. 2014, 85, 1141–1144. [CrossRef] 6. Hikita, T.; Koketsu, K.; Miyake, H. Variability of ground motion simulation due to aleatory uncertainty of source parameters. J. Jpn. Assoc. Earthq. Eng. 2020, 20, 21–34. [CrossRef] 7. Yang, L.; Xie, W.; Xu, W.; Ly, B.L. Generating drift-free, consistent, and perfectly spectrum-compatible time histories. Bull. Seismol. Soc. Am. 2019, 109, 1674–1690. [CrossRef] 8. Jayaram, N.; Lin, T.; Baker, J.W. A computationally efficient ground-motion selection algorithm for matching a target response spectrum mean and variance. Earthq. Spectra 2011, 27, 797–815. [CrossRef] 9. Cecini, D.; Palmeri, A. Spectrum-compatible accelerograms with harmonic wavelets. Comput. Struct. 2015, 147, 26–35. [CrossRef] 10. Zacchei, E.; Molina, J.L. Application of artificial accelerograms to estimating damage to dams using failure criteria. Int. J. Sci. Technol. 2020, 27, 2740–2751. 11. Boore, D.M. Simulation of ground motion using the Stochastic Method. Pure Appl. Geophys. 2003, 160, 635–676. [CrossRef] 12. Honda, R.; Ahmed, T. Design input motion synthesis considering the effect of uncertainty in structural and seismic parameters by feature indexes. J. Struct. Eng. 2011, 137, 391–400. [CrossRef] 13. Honda, R.; Khatri, P.P. Discrete analytic signal wavelet decomposition for phase localized in time-frequency domain for generation of stochastic signal with phase uncertainty. In Proceedings of the 15th Conference on Earthquake Engineering, Lisbon, Portugal, 24–28 September 2012. 14. Daubechies, I. Ten Lectures on Wavelets; Rutgers University and AT&T Bell Laboratories: Murray Hill, NJ, USA, 1992. 15. Fugal, D.L. Conceptual Wavelets in Digital Signal Processing; Space & Signals Technologies LLC: Spring Valley, CA, USA, 2009. 16. Kiban Kyoshin Net (KIK-NET). Available online: http://www.kik.bosai.go.jp (accessed on 14 September 2021). 17. Architectural Institute of Japan. Seismic Response Analysis and Design of Buildings Considering Dynamic Soil-Structure Interaction; Architectural Institute of Japan: Tokyo, Japan, 2006. (In Japanese) 18. The Open System for Earthquake Engineering Simulation. Available online: http://opensees.berkeley.edu (accessed on 14 September 2021).
http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png
Infrastructures
Multidisciplinary Digital Publishing Institute
http://www.deepdyve.com/lp/multidisciplinary-digital-publishing-institute/arbitrarily-oriented-phase-randomization-of-design-ground-motions-by-9u0N2VVRVH